Mathematics 7–10 · Year 9
Height from shadows: similar triangles with a metre stick
Measurement and space
The idea
At one moment the sun makes the same angle for every vertical object, so a metre stick and a tree with their shadows form similar triangles and the tree's height follows from one ratio.
Safety card
Hazards
- sun exposure
- looking at the sun
Controls
- hats, sunscreen, water
- never look at the sun; work with backs to the sun
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's excursion and playground supervision rules.
What you need
- 1 m stick or ruler held vertical (check with a plumb line)
- 30 m tape measure and chalk
- A sunny day with the shadow of the object falling on flat ground
How to do it
- Within the same two minutes measure the shadow of the vertical metre stick and the shadow of the object from its base to the shadow tip.
- Set up the proportion height of object / shadow of object = 1 m / shadow of stick and compute the height.
- Repeat 30 minutes later and compare: both shadows change, the ratio does not.
- Compare with the clinometer result for the same object if it was measured earlier.
What you should see
If the stick casts 1.25 m and the object casts 15.0 m, the height is 15.0 × 1.00/1.25 = 12.0 m. Half an hour later the shadows are longer or shorter but their ratio and the computed height agree with the first result within the measurement error of about 2 per cent (a 2 cm error on a 1.25 m stick shadow). The learner knows it worked when the two times of day give the same height within 0.3 m for a 12 m object and the result agrees with an independent clinometer measurement.
What changes
- What you change
- time of day (sun angle)
- What you measure
- computed height (should stay constant)
- What you keep the same
- stick held vertical
- both shadows measured within two minutes
- shadow tips read on flat ground
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The shadow is the same length as the object.
- The ratio changes with the time of day.
- Similar triangles need the same size to be compared.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Properties of geometrical figures A; page read 2026-09-22MA5-GEO-C-01
- Australian Curriculum v9AC9M9M03
Sources
The pages the author read to write this activity.